The Determinant of an Elliptic Sylvesteresque Matrix

• Published in: Symmetry, Integrability and Geometry: Methods and Applications
• Date: 2018
• Main Authors: Bhatnagar, G., Krattenthaler, C.
• Format: Article
• Language: English
• Published: Інститут математики НАН України 2018
• Online Access: https://nasplib.isofts.kiev.ua/handle/123456789/209520
• Journal Title: Digital Library of Periodicals of National Academy of Sciences of Ukraine
• Cite this: The Determinant of an Elliptic Sylvesteresque Matrix / G. Bhatnagar, C. Krattenthaler // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 13 назв. — англ.

We evaluate the determinant of a matrix whose entries are elliptic hypergeometric terms and whose form is reminiscent of Sylvester matrices. A hypergeometric determinant evaluation of a matrix of this type has appeared in the context of approximation theory, in the work of Feng, Krattenthaler, and Xu. Our determinant evaluation is an elliptic extension of their evaluation, which has two additional parameters (in addition to the base q and nome p found in elliptic hypergeometric terms). We also extend the evaluation to a formula transforming an elliptic determinant into a multiple of another elliptic determinant. This transformation has two further parameters. The proofs of the determinant evaluation and the transformation formula require an elliptic determinant lemma due to Warnaar, and the application of two Cn elliptic formulas that extend Frenkel and Turaev's ₁₀V₉ summation formula and ₁₂V₁₁ transformation formula, results due to Warnaar, Rosengren, Rains, and Coskun and Gustafson.